In the next post of our mathematics in action series, we look at tessellations of the plane. The most familiar of these are the three regular tilings, using tiles that are regular triangles, squares, or (as below) hexagons.
Alternatively, the regular tilings can be extended by mixing different kinds of regular polygon. Of particular interest are the eight semiregular tilings, in which the tiles all meet edge-to-edge, and each vertex is equivalent to each other vertex (i.e. each vertex can be mapped to each other vertex through rotations, reflections, translations, or glide reflections). Here is one of the eight:
Because of the high level of symmetry, an exhaustive list of the 11 regular and semiregular tilings can be made by considering all possible meetings of polygons at a vertex, such as these two:
Penrose tilings, discovered by Roger Penrose in 1974, loosen the regularity and symmetry conditions, while still using a fixed number of kinds of tile, and while still being “almost” symmetrical. In the image below, Penrose is standing on a Penrose tiling. His 1974 discovery goes to show that fairly simple mathematical truths can still be discovered today.
We are all familiar with dice of various kinds. With fair dice, each number will come up with equal probability, regardless of how the die is rolled. This requires a degree of symmetry – we want a die to be a polyhedron where all the faces are equivalent. The obvious candidates are therefore the five Platonic solids, in which not only the faces, but also the edges and vertices are equivalent. The Platonic solids give us the common d4, d6, d8, d12, and d20 dice:
However, the Platonic solids are more symmetrical than necessary for the job. A tetragonal disphenoid, for example, makes a very good d4:
A tetragonal disphenoid makes an alternative d4 – photo by “Traitor”
What is required is that a die be isohedral (also called “face-transitive”). Each face should be equivalent. Specifically, for any numbers A and B, given the die with A on top, there should be a series of rotations and reflections that make the die look like the starting position, but with B on top. This rules out shapes like the gyroelongated square bipyramid, where all the faces are equilateral triangles, but the triangles are not equivalent (the “end” triangles differ from the “middle” triangles):
A gyroelongated square bipyramid does not make a fair die – photo by Andrew Kepert
We also want a die to be convex, so that it can land on its faces. Stellated polyhedra are excluded:
Trapezohedra satisfy this “convex and isohedral” rule, and the pentagonal trapezohedron is commonly used as a d10 (see the picture above). Trapezohedra work best with 10, 14, 18, … sides, since then pairs of faces can be parallel, and there can be an unambiguous “top” number. The cube can be seen as a special case of a trapezohedron.
For 12, 16, 20, …. sides, bipyramids make good dice (the octahedron is a special case of a bipyramid):
A bipyramid makes a good d16 – photo by “Traitor”
These are not the only shapes satisfying the definition, however. The 13 Catalan solids also satisfy it, and some of them make good candidates for dice. For example, the deltoidal icositetrahedron and the tetrakis hexahedron are both good candidates for d24:
The deltoidal icositetrahedron and tetrakis hexahedron are alternatives for d24 – photos by Jacqueline de Swart (left) and “Traitor” (right)
Some Catalan solids, like the pentagonal icositetrahedron, are unsuitable as dice because there is no unambiguous “top” number. On the other hand, there are some additional variations that are isohedral, like the hexakis tetrahedron.
For more on this subject, see Alea Kybos’ impressive dice page.
I find paperweights useful while working, and all the more so if they are appropriate in some way. Here is part of my collection.
On the left is a marble cuboctahedron. Since the edges of this shape form a symmetric graph (all 24 edges are equivalent), it made a nice example while I was working on my 2004 paper “Network Robustness and Graph Topology.” For example, the cuboctahedron is one of the Cayley graphs for the alternating group A4.
The frog on the right, on the other hand, feels right at home with some current work I’m doing on amphibian species distribution modelling.
“The chief forms of beauty are order and symmetry and definiteness, and these are especially manifest in the mathematical sciences” (τοῦ δὲ καλοῦ μέγιστα εἴδη τάξις καὶ συμμετρία καὶ τὸ ὡρισμένον, ἃ μάλιστα δεικνύουσιν αἱ μαθηματικαὶ ἐπιστῆμαι) – Aristotle, Metaphysics, Book 13 (Mu), Section 3, my translation.
“Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection” – Hermann Weyl, Symmetry, 1952, Princeton University Press, p. 5.
“Regularity is successive symmetry; there is no reason, therefore, to be astonished that the forms of equilibrium are often symmetrical and regular” – Ernst Mach, The Science Of Mechanics, 1919 edition, p. 395.
Bottom left image derived from a public domain photo by Vinoo202.
Continuing the theme of the Foster Census of cubic symmetric graphs (networks) from my last post, here is the graph F96B from that census (click on the picture for a larger image). This beautifully symmetrical graph has 96 nodes (all equivalent) and 144 edges (also all equivalent). The image does not do justice to the symmetry of this graph.
Graph F96B is bipartite: the two classes of node are coloured red and blue. It has diameter 7 and girth 8. One of the 8-rings within the graph is highlighted in orange. The graph can be expressed in LCF notation as [−45, −33, −15, 45, −39, −21, −45, 39, 21, 45, −15, 15, −45, 39, −39, 45, 33, 27, −45, 15, −27, 45, −39, 39]4.